 Solution

The derivatives of the Lagrangian density are ${\mathcal{L}}_{\varphi }=0$, ${\mathcal{L}}_{{\varphi }_t}={\varphi }_t$, and ${\mathcal{L}}_{{\varphi }_x}=-{\varphi }_x$. The Euler-Lagrange field equation is thereforeThis is the one-dimensional wave equation with wave speed $c_s=\sqrt{\kappa /\rho }=1$. Both signs of propagation are allowed: $\varphi (x,t)=F(x-t)+H(x+t)$ for sufficiently regular profiles.